Monday, September 15, 2008

Basic Instinct: Math

Natalie Angier has an article in today's NYT linking advanced math skills to our gut feeling for approximating things. There may be such a correlation, but I'm not impressed by this example:

When mathematicians and physicists are left alone in a room, one of the games they’ll play is called a Fermi problem, in which they try to figure out the approximate answer to an arbitrary problem,” said Rebecca Saxe, a cognitive neuroscientist at the Massachusetts Institute of Technology who is married to a physicist. “They’ll ask, how many piano tuners are there in Chicago, or what contribution to the ocean’s temperature do fish make, and they’ll try to come up with a plausible answer.”

“What this suggests to me,” she added, “is that the people whom we think of as being the most involved in the symbolic part of math intuitively know that they have to practice those other, nonsymbolic, approximating skills

I'm pretty sure physicists don't use gut instinct to solve Fermi problems - at least I don't. What I, and I'm pretty sure other physicists use instead is a combination of analysis and calculation to come up with an approximation. Analysis breaks the problem in to pieces that you can relate to things you know.

One classic Fermi problem: how many snowflakes fall in a typical New England snowstorm?

How do you attack that? I would first try to come up with a volume estimate. I've never been in a New England snowstorm, but I will guess that it produces an average snow depth of say 8 inches, or 20 cm. How big is New England? It looks kind of small on the map in my head (looking at a real map would be cheating) - maybe 15% of the whole country, which I seem to recall is about 3 million square miles or roughly (quick calculation) 7.5 million sq km. 15 % of that (quick mental calculation) is about 1.1 million km^2 = (another qc) 1.1 x 10^16 cm^2 so total volume of our standard snowstorm is about 2 x 10^17 cm^3.

Snow varies between 10% and about 90% water, I think (the rest being air), but NE probably gets the heavy wet stuff, so lets say 50%, so the weight is about 10^17 gm.

Now all we have to do is figure out the weight of a typical snowflake and divide by that! I have seen a lot of snowflakes, and they vary over many orders of magnitude, but I will guess that a typical lands on my nose snowflake has a volume of about 0.1 cm^3 of which all but about 0.02 cm^3 is air.

So (dividing by 0.02, i.e. multiplying by 50) gives 5 x 10^18 snowflakes. If you've counted, please check my work. The part of the calculation in which I have the least confidence is the weight of the snowflake part. It doesn't snow much in southern New Mexico.

The point though, is that this is all analysis, memory, and calculation. The only gut estimates were the % NE forms of the US, and the size of the snowflake.

If anybody else does these problems a lot differently, I would like to hear about it.